EM-based channel estimation in OFDM systems with phase noise

EM-based channel estimation in OFDM systems with phase noise Rodrigo Carvajal, Juan C. Agüero, Boris I. Godoy, Graham C. Goodwin. Centre for Complex Dynamic Systems and Control (CDSC) The University of Newcastle, Australia. 4th of January, 2012

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Outline

1

Problem of interest

2

Introduction

3

OFDM Systems Phase distortion in OFDM systems System model

4

Channel Estimation in OFDM systems with phase noise

5

Numerical Examples

6

Extensions of our work

7

Conclusions

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Problem of interest General model with phase noise: φk+1 = φk + vk+1 , xk+1 = xk , ˜ + ηk, rk = ejφk (eTk H)x

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

ηk ∼ N(0, ση2 )

EM-based channel estimation in OFDM systems with phase noise

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Problem of interest General model with phase noise: φk+1 = φk + vk+1 , xk+1 = xk , ˜ + ηk, rk = ejφk (eTk H)x

ηk ∼ N(0, ση2 )

We propose to solve the estimation problem using Maximum likelihood. Difficulties: I I

Clearly define parameters and variables. R Hidden variables z, i.e. p(r|θ ) = p(r, z|θ )dz.

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Problem of interest General model with phase noise: φk+1 = φk + vk+1 , xk+1 = xk , ˜ + ηk, rk = ejφk (eTk H)x

ηk ∼ N(0, ση2 )

We propose to solve the estimation problem using Maximum likelihood. Difficulties: I I

Clearly define parameters and variables. R Hidden variables z, i.e. p(r|θ ) = p(r, z|θ )dz.

 Possibilities

 Difficulties

 h, x: random variables

 Nonlinearities

 h, x: constant parameters

 Singular probability density function

 h: random variables , x: constant parameter  h: constant parameter , x: random variable R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

 Number of data points (NC ) smaller than the number of parameters

EM-based channel estimation in OFDM systems with phase noise

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Problem of interest General model with phase noise: φk+1 = φk + vk+1 , xk+1 = xk , ˜ + ηk, rk = ejφk (eTk H)x

ηk ∼ N(0, ση2 )

We propose to solve the estimation problem using Maximum likelihood. Difficulties: I I

Clearly define parameters and variables. R Hidden variables z, i.e. p(r|θ ) = p(r, z|θ )dz.

 Possibilities

 Difficulties

 h, x: random variables

 Nonlinearities  Singular probability density function

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Problem of interest General model with phase noise: φk+1 = φk + vk+1 , xk+1 = xk , ˜ + ηk, rk = ejφk (eTk H)x

ηk ∼ N(0, ση2 )

We propose to solve the estimation problem using Maximum likelihood. Difficulties: I I

Clearly define parameters and variables. R Hidden variables z, i.e. p(r|θ ) = p(r, z|θ )dz.

 Possibilities

 Difficulties  Nonlinearities

 h, x: constant parameters  Number of data points (NC ) smaller than the number of parameters R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Problem of interest General model with phase noise: φk+1 = φk + vk+1 , xk+1 = xk , ˜ + ηk, rk = ejφk (eTk H)x

ηk ∼ N(0, ση2 )

We propose to solve the estimation problem using Maximum likelihood. Difficulties: I I

Clearly define parameters and variables. R Hidden variables z, i.e. p(r|θ ) = p(r, z|θ )dz.

 Possibilities

 Difficulties  Nonlinearities

 h: random variables , x: constant parameter

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

 Singular probability density function  Number of data points (NC ) smaller than the number of parameters

EM-based channel estimation in OFDM systems with phase noise

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Problem of interest General model with phase noise: φk+1 = φk + vk+1 , xk+1 = xk , ˜ + ηk, rk = ejφk (eTk H)x

ηk ∼ N(0, ση2 )

We propose to solve the estimation problem using Maximum likelihood. Difficulties: I I

Clearly define parameters and variables. R Hidden variables z, i.e. p(r|θ ) = p(r, z|θ )dz.

 Possibilities

 Difficulties  Nonlinearities  Singular probability density function

 h: constant parameter , x: random variable R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Introduction The EM algorithm has previously utilized in OFDM systems parameter estimation1 The estimation of multiple parameters in OFDM systems with phase noise has been addressed previously in the literature2

1

R. Mo, et. al, “An EM-based semiblind joint channel and frequency offset estimator for OFDM systems over frequency selective fading channels,” IEEE Trans. Veh. Technol., vol. 57, no. 5, pp.3275–3282, Sep. 2008.

2

F. Septier et. al, “Monte-Carlo methods for channel, phase noise, and frequency offset estimation with unknown noise variances in OFDM systems,” IEEE Trans. Signal Process., vol. 56, no. 8, pp. 3613–3626, Aug. 2008. R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Introduction

→ In our approach, the channel is regarded as a constant parameter. → We exploit the linear and Gaussian structure associated with the transmitted signal. → We obtain expressions that consider different levels of training. → We study the joint estimation of PHN bandwidth and CIR, and the effect that the associated estimation errors have on the CIR. → We show that inaccurate PHN bandwidth estimation introduces errors in the estimation of the CIR when the number of subcarriers is low. 1

R. Mo, et. al, “An EM-based semiblind joint channel and frequency offset estimator for OFDM systems over frequency selective fading channels,” IEEE Trans. Veh. Technol., vol. 57, no. 5, pp.3275–3282, Sep. 2008.

2

F. Septier et. al, “Monte-Carlo methods for channel, phase noise, and frequency offset estimation with unknown noise variances in OFDM systems,” IEEE Trans. Signal Process., vol. 56, no. 8, pp. 3613–3626, Aug. 2008. R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Outline

1

Problem of interest

2

Introduction

3

OFDM Systems Phase distortion in OFDM systems System model

4

Channel Estimation in OFDM systems with phase noise

5

Numerical Examples

6

Extensions of our work

7

Conclusions

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Phase distortion in OFDM systems

Carrier frequency offset Doppler effect

⇒ frequency variations from the nominal frequency.

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Phase distortion in OFDM systems

Carrier frequency offset Doppler effect

⇒ frequency variations from the nominal frequency.

Phase noise → Arises due to random fluctuations of the Rx and Tx oscillators. → Its behaviour depends on the type of oscillator used, i.e., PLLs or free running oscillators.

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Phase distortion in OFDM systems

Carrier frequency offset Doppler effect

⇒ frequency variations from the nominal frequency.

Phase noise → Arises due to random fluctuations of the Rx and Tx oscillators. → Its behaviour depends on the type of oscillator used, i.e., PLLs or free running oscillators. These frequency synchronization errors compromise the orthogonality of the subcarriers and create intercarrier interference (ICI)

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

6 / 21

Phase distortion in OFDM systems

Carrier frequency offset Doppler effect

⇒ frequency variations from the nominal frequency.

Phase noise → Arises due to random fluctuations of the Rx and Tx oscillators. → Its behaviour depends on the type of oscillator used, i.e., PLLs or free running oscillators. These frequency synchronization errors compromise the orthogonality of the subcarriers and create intercarrier interference (ICI)

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

6 / 21

Phase distortion in OFDM systems

Carrier frequency offset Doppler effect

⇒ frequency variations from the nominal frequency.

Phase noise → Arises due to random fluctuations of the Rx and Tx oscillators. → Its behaviour depends on the type of oscillator used, i.e., PLLs or free running oscillators. These frequency synchronization errors compromise the orthogonality of the subcarriers and create intercarrier interference (ICI)

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

6 / 21

Outline

1

Problem of interest

2

Introduction

3

OFDM Systems Phase distortion in OFDM systems System model

4

Channel Estimation in OFDM systems with phase noise

5

Numerical Examples

6

Extensions of our work

7

Conclusions

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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System Model

Phase Noise → In free-running oscillators, PHN is modelled as a continuous Brownian motion process: φk+1 = φk + vk+1 , where: - vk : i.i.d zero-mean Gaussian variable with variance σv2 = 2πβ T/NC . - β : PHN bandwidth. - T symbol duration (1/T is the symbol rate).

Carrier Frequency Offset → CFO can be modelled as a diagonal matrix Cε = e 0, 1, . . . , NC − 1.

jdiag



2πεk NC



, with k =

→ ε is the normalized frequency offset (|ε| ≤ 1/2). R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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System Model We assume that the cyclic prefix has been successfully removed.

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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System Model We assume that the cyclic prefix has been successfully removed. The transmitted signal x can be expressed as a linear combination of a deterministic signal (known) and a stochastic signal (unkwnown), as x = (¯xk + x˜ ). Then, we have: φk+1 = φk + vk+1 , vk ∼ N(0, 2πβ T/NC ) x¯ k+1 = x¯ k , ˜ xk + x˜ ) + η k , ηk ∼ N(0, ση2 ) rk = ejψk (eTk H)(¯ where I I I I I

˜ is the (circulant) channel matrix, H ψk = φk + 2πkε NC , x¯ k is the (unknown) stochastic part of x, x˜ is the (known) training part of x, and ek is the kth column of the identity matrix.

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Outline

1

Problem of interest

2

Introduction

3

OFDM Systems Phase distortion in OFDM systems System model

4

Channel Estimation in OFDM systems with phase noise

5

Numerical Examples

6

Extensions of our work

7

Conclusions

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Channel Estimation in OFDM systems with phase noise We assume that there is no a priori knowledge of the parameters to be estimated.

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Channel Estimation in OFDM systems with phase noise We assume that there is no a priori knowledge of the parameters to be estimated. Then, we solve our estimation problem using Maximum Likelihood.

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Channel Estimation in OFDM systems with phase noise We assume that there is no a priori knowledge of the parameters to be estimated. Then, we solve our estimation problem using Maximum Likelihood. The estimation problem requires the maximization of θˆ = arg max l(θ ), θ

where I I I

θ = [hT , (β T)−1 ], l(θ ) = log {p(r | θ )}, p(r | θ ) is the marginal pdf of the received data.

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Channel Estimation in OFDM systems with phase noise We assume that there is no a priori knowledge of the parameters to be estimated. Then, we solve our estimation problem using Maximum Likelihood. The estimation problem requires the maximization of θˆ = arg max l(θ ), θ

where I I I

θ = [hT , (β T)−1 ], l(θ ) = log {p(r | θ )}, p(r | θ ) is the marginal pdf of the received data.

Difficulties → The system is nonlinear (due to the exponential ejφk ). → There is no measurement of the phase noise, φ0:NC −1 . R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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The Expectation-Maximization (EM) algorithm EM is an iterative algorithm to obtain the ML estimate3 . Unknown signals are treated as hidden variables3 .

3

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B, vol. 39, no. 1, pp. 1–38, 1977.

4

T. Schön, F. Gustaffson, and P.-J. Nordlund, “Marginalized particle filters for mixed linear/nonlinear state-space models,” IEEE Trans. Signal Process., vol. 53, no. 7, pp. 2279–2289, July 2005. R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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The Expectation-Maximization (EM) algorithm EM is an iterative algorithm to obtain the ML estimate3 . Unknown signals are treated as hidden variables3 . E-step: Compute the expected value Q(θ , θˆ k ) = E{ log[p(¯x, φ , r|θ )]|r, θˆ k } M-step: Obtain a new estimate by maximizing the function Q, as θˆ k+1 = arg max Q(θ , θˆ k ) θ

3

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B, vol. 39, no. 1, pp. 1–38, 1977.

4

T. Schön, F. Gustaffson, and P.-J. Nordlund, “Marginalized particle filters for mixed linear/nonlinear state-space models,” IEEE Trans. Signal Process., vol. 53, no. 7, pp. 2279–2289, July 2005. R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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The Expectation-Maximization (EM) algorithm EM is an iterative algorithm to obtain the ML estimate3 . Unknown signals are treated as hidden variables3 . l (θ)

E-step: Compute the expected value Q(θ , θˆ k ) = E{ log[p(¯x, φ , r|θ )]|r, θˆ k } M-step: Obtain a new estimate by maximizing the function Q, as θˆ k+1 = arg max Q(θ , θˆ k ) θ

Q(θ , θ̂ k +1 )

Q (θ , θ̂ k )

θ̂ k

θ̂ k +1

θ̂ k +2

θ

Graphical representation of the EM algorithm

The implementation of the EM algorithm involves the computation of the probability density function p(¯x, φ |r, θ ). 3

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B, vol. 39, no. 1, pp. 1–38, 1977.

4

T. Schön, F. Gustaffson, and P.-J. Nordlund, “Marginalized particle filters for mixed linear/nonlinear state-space models,” IEEE Trans. Signal Process., vol. 53, no. 7, pp. 2279–2289, July 2005. R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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The Expectation-Maximization (EM) algorithm EM is an iterative algorithm to obtain the ML estimate3 . Unknown signals are treated as hidden variables3 . l (θ)

E-step: Compute the expected value Q(θ , θˆ k ) = E{ log[p(¯x, φ , r|θ )]|r, θˆ k }

Q(θ , θ̂ k +1 )

Q (θ , θ̂ k )

M-step: Obtain a new estimate by maximizing the function Q, as θˆ k+1 = arg max Q(θ , θˆ k ) θ

θ̂ k

θ̂ k +1

θ̂ k +2

θ

Graphical representation of the EM algorithm

We obtain p(¯x, φ |r, θ ) = p(¯x|φ , r, θ ) p(φ |r, θ ) , by utilizing the Marginalized | {z } | {z } Kalman filter Particle filter

particle filter4 . 3

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B, vol. 39, no. 1, pp. 1–38, 1977.

4

T. Schön, F. Gustaffson, and P.-J. Nordlund, “Marginalized particle filters for mixed linear/nonlinear state-space models,” IEEE Trans. Signal Process., vol. 53, no. 7, pp. 2279–2289, July 2005. R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Outline

1

Problem of interest

2

Introduction

3

OFDM Systems Phase distortion in OFDM systems System model

4

Channel Estimation in OFDM systems with phase noise

5

Numerical Examples

6

Extensions of our work

7

Conclusions

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Numerical Examples

Simulation setup: The signal x is considered known (training). A Rayleigh fading channel with L = 4 taps, (β T)−1 = 1000 (β T = 0.001) Known ε = 0.2537, ση2 = 0.01, and SNR = 10[dB]. The number of particles used in the particle smoother is 100 (constant), and the number of iterations of the EM algorithm is 150. The vector parameter to estimate is θ = [hT (β T)−1 ]T . In a recent paper, it was shown that the PHN bandwidth cannot be accurately estimated5 . Hence, several initial guess are considered for (β T)−1 .

5 R. Carvajal, J. C. Agüero, B. I. Godoy, and G. C. Goodwin, “On the accuracy of phase noise bandwidth estimation in OFDM systems,” in Proc. IEEE Int. Work. Signal Process. Adv. Wireless Commun. (SPAWC) pp. 246–250, San Francisco, USA, 26–29 June 2011. R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Numerical Examples NC = 64

4.5

3 (βT)−1 known 0

4

(βT)−1 = 1×10−3(βT)−1 ini 0 (βT)−1 ini (βT)−1 ini (βT)−1 ini

3 2.5

= = =

1

40(βT)−1 0 1×103(βT)−1 0 1.25(βT)−1 0

Phase [rad]

Magnitude [−]

2

(βT)−1 = 0.025(βT)−1 ini 0

3.5

True C.F.R. 2

0

−1

(βT)ini = 0.025(βT)0 −1

−3

−1

(βT)ini = 1×10 (βT)0 −2

1.5

(βT)−1 known 0 −1

−1

1

−3

0.5 0

−4 0

−1 (βT)ini (βT)−1 ini −1 (βT)ini

= = =

−1 40(βT)0 1×103(βT)−1 0 −1 1.25(βT)0

True C.F.R 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Normalized frequency × π [rad/sample]

1.8

2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Normalized frequency × π [rad/sample]

Frequency response of the estimated channel with different initial guesses for the PHN bandwidth. NC = 64, (β T)−1 0 = 1000.

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Numerical Examples NC = 256

4.5

3 (βT)−1 known 0 (βT)−1 ini (βT)−1 ini (βT)−1 ini (βT)−1 ini (βT)−1 ini

Magnitude [−]

3.5 3 2.5

= = = = =

0.025(βT)−1 0 1×10−3(βT)−1 0 −1 40(βT)0 3 −1 1×10 (βT)0 0.8(βT)−1 0

2

1

Phase [rad]

4

True C.F.R 2

0

−1

(βT)ini = 0.025(βT)0 −1

−3

−1

(βT)ini = 1×10 (βT)0 −2

1.5

(βT)−1 known 0 −1

−1

1

−3

0.5 0

−4 0

−1 (βT)ini (βT)−1 ini −1 (βT)ini

= = =

−1 40(βT)0 1×103(βT)−1 0 −1 0.8(βT)0

True C.F.R 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Normalized frequency × π [rad/sample]

1.8

2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Normalized frequency × π [rad/sample]

Frequency response of the estimated channel with different initial guesses for the PHN bandwidth. NC = 256, (β T)−1 0 = 1000.

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Outline

1

Problem of interest

2

Introduction

3

OFDM Systems Phase distortion in OFDM systems System model

4

Channel Estimation in OFDM systems with phase noise

5

Numerical Examples

6

Extensions of our work

7

Conclusions

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Extensions of our work

Submitted to IEEE Trans. on Veh. Technol.

Extension to the estimation of CFO, ε, and channel noise variance, ση2 . Extension to proper and improper signals (for different modulation schemes, such as BPSK, GMSK). Analysis of the impact of different training levels on the overall parameter estimation. Analytic expression for Cramer Rao Lower Bound of phase noise bandwidth (expressed as (β T)−1 ).

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Outline

1

Problem of interest

2

Introduction

3

OFDM Systems Phase distortion in OFDM systems System model

4

Channel Estimation in OFDM systems with phase noise

5

Numerical Examples

6

Extensions of our work

7

Conclusions

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

EM-based channel estimation in OFDM systems with phase noise

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Conclusions

→ We have exploited the linear and Gaussian structure associated with the transmitted signal. → Maximum likelihood estimation of the channel impulse response (CIR) can be successfully performed in OFDM systems, based on particle smoothing and the EM algorithm. → The impact of the inaccurate PHN bandwidth estimation on the estimation of CIR is negligible when the number of subcarriers is relatively high (e.g. 256 or more). → For a small number of subcarriers, inaccurate estimation of CIR can potentially have a significant impact on the estimation of the received signal and hence lead to an increase in the bit error rate.

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

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Questions ?

R. Carvajal, J. C. Agüero, B. I. Godoy, G. C. Goodwin.

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